Here's some code that does basically what apl is suggesting (in the spirit of getting you started, but leaving some things for you to do to enhance the learning experience):

#! /usr/bin/perl

use strict;
use warnings;

my @file1 = (
             'A101 something 123',
             'A102 else 456',
             'A103 it 789',
             'A104 the 012'
            );

my @file2 = (
             'A102 else 456',
             'A103 it 789'
            );

my %prod_id;

foreach my $line ( @file1 ) {
  my( $num, $name, $id ) = (split /\s+/, $line);
  $prod_id{$num} = $id;
}

foreach my $line ( @file2 ) {
  my( $num, $name ) = (split /\s+/, $line);
  print "$num\t$name\t$prod_id{$num}\n";
}
[download]

There is still a lot of work that you need to do with this. For example, to make it easy on myself, I take sample data from arrays, not files (you will need to fix that part). I also took some shortcuts by (1) always ignoring ProductName, (2) always assuming that the relevant ProductID exists as a key in the hash (this really needs to be checked at the point where I use $prod_id{$num} above).

As I said, this is not offered as a complete solution, rather as something to get you started.